Controlled homotopy equivalences and structure sets of manifolds

Abstract:

For a closed topological $n$–manifold $K$ and a map $p:K\to B$ inducing an isomorphism $\pi _{1}(K)\to \pi _{1}(B)$, there is a canonically defined morphism $b:H_{n+1}(B,K,\mathbb {L} )\to \mathcal {S}(K)$, where $\mathbb {L}$ is the periodic simply connected surgery spectrum and $\mathcal {S}(K)$ is the topological structure set. We construct a refinement $a:H_{n+1}^{+}(B,K,\mathbb {L} )\to \mathcal {S}_{\varepsilon ,\delta }(K)$ in the case when $p$ is $UV^{1}$, and we show that $a$ is bijective if $B$ is a finite-dimensional compact metric ANR. Here, $H_{n+1}^{+}(B,K,\mathbb {L} )\subset H_{n+1}(B,K,\mathbb {L} )$, and $\mathcal {S}_{\varepsilon ,\delta }(K)$ is the controlled structure set. We show that the Pedersen-Quinn-Ranicki controlled surgery sequence is equivalent to the exact $\mathbb {L}$-homology sequence of the map $p:K \to B$, i.e. that \begin{equation*}H_{n+1}(B,\mathbb {L} )\to H_{n+1}^{+}(B,K,\mathbb {L} )\to H_{n}(K,\mathbb {L} ^{+})\to H_{n}(B,\mathbb {L} ), \ \mathbb {L} ^{+}\to \mathbb {L}, \end{equation*} is the connected covering spectrum of $\mathbb {L}$. By taking for $B$ various stages of the Postnikov tower of $K$, one obtains an interesting filtration of the controlled structure set.